Predicates as processes: Linear implication is a branching-time causality-preserving precongruence

We summarise recent advances in the predicates-as-processes paradigm. In particular, we note the non-commutative logic MAV1 featuring choice and a novel dual pair of nominal quantifiers. For fragments of MAV1, we have shown the soundness of implication with respect to both weak simulation and pomset trace inclusion; hence implication is a branching-time causality-preserving precongruence. The system is sufficiently expressive to soundly embed finite fragments of process models not limited to the π-calculus, session type systems and workflow modelling languages. 1 A roadmap for the predicates-as-processes paradigm There are striking formal connections between linear logic and interaction and concurrency in process calculi. Attempts to formalise such connections could be classified along two major thrusts: the proofs-as-process [1, 4] and predicates-as-processes paradigms. We follow the predicates-as-processes paradigm where processes are embedded directly as predicates in a logical system. Early work [12], considered linear logic as the target system. However, such embeddings tended to encoded the semantics of sequentiality as a theory rather than a logical primitive; hence implication does not directly give rise to a preorder over processes. The calculus of structures [8], a generalisation of the sequent calculus, is sufficiently expressive to extend linear logic with a non-commutative operator suitable for modelling sequentiality as a logical primitive. Preliminary work [3], established that linear implication in proof system BV is sound (but not complete) with respect to completed trace inclusion for a fragment of CCS with only prefix and parallel composition. This preliminary investigation can be tightened according to the following agenda: • Where is linear implication precisely situated in the spectrum of process preorders? • Can expressive process calculi be soundly embedded in extensions of BV? • Given that there is a strong objective justification for this process preorder (cut elimination); are there also compelling applications for this process preorder? Figure 1 elaborates on the first question above. In Fig. 1, process preorders are divided along two axis: the linear-time/branching-time axis, and the interleaving/causality-preserving axis [13]. At the top of Fig. 1 is trace inclusion, defined by subset inclusion over the set of all traces of a process. Trace inclusion is widely considered to be the coarsest preorder over processes; hence preorders should be sound with respect to trace inclusion, as indicated by the arrows in the figure. Along the lineartime/branching-time axis, trace inclusion can be refined by various weak simulations which have finer properties regarding the distributivity of non-deterministic choice, indicated by ⊕ in this work. In the other direction, along the interleaving/causality-preserving axis, models such as pomset traces [7] reserve causal relationships between events; ensuring, for example, that, unlike trace inclusion and weak simulation, we have action refinement and no autoconcurrency [2]. In recent work, we observe that linear implication is both branching-time and causality-preserving; hence is situated at the bottom of Fig. 1. Linear Implication is a Branching-time Causality-preserving Precongruence Horne and Tiu interleaving autoconcurrency tr(a ; a) = tr(a ‖ a) traces tr(P) ⊆ tr(Q) linear time tr(a ; (b ⊕ c)) = tr((a ; b) ⊕ (a ; c)) weak simulation P Q 66 pomset traces ideals(P) ⊆ ideals(Q) hh branching time a ; (b ⊕ c) “ (a ; b) ⊕ (a ; c) (a ; b) ⊕ (a ; c)( a ; (b ⊕ c) implication ` P( Q Bruscoli (2002) OO Horne, Mauw and Tiu (2017) 66 Horne and Tiu (submitted) gg causality-preserving no autoconcurrency a ‖ a “ a ; a a ; a( a ‖ a Figure 1: A roadmap situating implication in the spectrum of process preorders. 2 A first-order non-commutative logic with nominals We recall a first-order extension of BV, called MAV1 [11], featuring a novel de Morgan dual pair of nominal quantifiers “new” and “wen” introduced to model private names as featured in the π-calculus. The syntax is presented in Fig. 2 and the proof system is defined by a term-rewriting system modulo an equational theory, where rules can be applied deep within any context. The semantics ensures that seq is non-commutative, whereas other operators are commutative. P F I (unit) α (atom) α (co-atom) ∀x.P (all) Иx.P (new) Эx.P (wen) ∃x.P (some) P & P (with) P ⊕ P (plus) P ‖ P (par) P ⊗ P (times) P ; P (seq) Figure 2: MAV1 syntax. Linear negation is defined by a function over predicates such that the first-order quantifiers ∃ and ∀ are de Morgan dual, as are the nominal quantifiers И and Э, the multiplicatives ⊗ and ‖, the additives & and ⊕, and atoms α and α. The non-commutative operator seq and unit are self-dual in the sense that they are their own de Morgan duals. Linear implication P( Q is defined in terms of linear negation and par as P ‖ Q. A derivation is a sequence of zero or more rewrites and a proof of P, is a derivation rewriting P to the unit. When such a derivation exists, we say that P is provable, and write ` P. A generalisation of cut elimination holds for MAV1 [9, 11] as follows. Theorem 2.1 (Horne et al. (2016)). If ` C { P ⊗ P } then ` C{ I }. Cut elimination is the corner stone of a proof calculus. A corollary of significance here is that linear implication defines a precongruence. For the sub-calculus, MAV, without quantifiers, the following soundness result has been established [10]. Theorem 2.2 (Horne, Mauw and Tiu (2017)). For series-parallel processes with choice, if ` P ( Q then the pomset traces of P are included in the pomset traces of Q. In the above, pomset traces are defined in terms of ideals given by certain graph homomorphisms [7]. The result was stated for a workflow model called an attack tree, but covers related process models [5] and should extend easily to pomset trace semantics for CCS and the π-calculus.